Open Access

A characterization of chaotic order

Journal of Inequalities and Applications20062006:79123

DOI: 10.1155/JIA/2006/79123

Received: 15 November 2005

Accepted: 4 January 2006

Published: 8 June 2006

Abstract

The chaotic order https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq1_HTML.gif among positive invertible operators https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq2_HTML.gif on a Hilbert space is introduced by https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq3_HTML.gif . Using Uchiyama's method and Furuta's Kantorovich-type inequality, we will point out that https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq4_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq5_HTML.gif holds for any https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq6_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq7_HTML.gif is any fixed positive number. On the other hand, for any fixed https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq8_HTML.gif , we also show that there exist positive invertible operators https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq9_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq10_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq11_HTML.gif holds for any https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq12_HTML.gif , but https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F79123/MediaObjects/13660_2005_Article_1639_IEq13_HTML.gif is not valid.

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Authors’ Affiliations

(1)
Department of Mathematics, Henan Normal University

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Copyright

© C. Yang and F. Gao 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.